This diagram indicates that
in certain specific non-Hermitian-Hamiltonian models
one has to separate
the probabilistic
physical predictions formulated in \({\cal H}_{phys}\) 
from the explicit calculations
performed, much more efficiently, in
\({\cal H}_{math}\) .
The
space \({\cal L}\)  and the SP
Hamiltonian \(\mathfrak {h}\) 
have to be abandoned
and replaced by
the doublet \({\cal H}_{math/phys}\) 
and by the preconditioned
Hamiltonian \(H\) , respectively.
A very general crypto-Hermitian SP (CHSP) of review [1]}
is born.
